Anomaly Inflow and the $\eta$-Invariant

TL;DR

This paper provides a nonperturbative framework for understanding fermion anomalies using the $ a$-invariant, extending the anomaly inflow concept to include global anomalies and justifying previous theoretical expectations.

Contribution

It introduces a formula involving the $ a$-invariant for anomaly inflow, unifying perturbative and nonperturbative anomalies in a rigorous, cobordism-invariant manner.

Findings

The $ a$-invariant describes anomaly inflow nonperturbatively.

The formula generalizes anomaly inflow to include global anomalies.

The approach is consistent with the Dai-Freed theorem.

Abstract

Perturbative fermion anomalies in spacetime dimension $d$ have a well-known relation to Chern-Simons functions in dimension $D=d+1$. This relationship is manifested in a beautiful way in "anomaly inflow" from the bulk of a system to its boundary. Along with perturbative anomalies, fermions also have global or nonperturbative anomalies, which can be incorporated by using the $\eta$-invariant of Atiyah, Patodi, and Singer instead of the Chern-Simons function. Here we give a nonperturbative description of anomaly inflow, involving the $\eta$-invariant. This formula has been expected in the past based on the Dai-Freed theorem, but has not been fully justified. It leads to a general description of perturbative and nonperturbative fermion anomalies in $d$ dimensions in terms of an $\eta$-invariant in $D$ dimensions. This $\eta$-invariant is a cobordism invariant whenever perturbative anomalies cancel.